Question

In how many different ways can all the letters of the word zookeeper be arranged?

Answer 1

An infinite amount seeing as you can have infinite of each number

Answer 2

The 9! ways of arranging the letters in ZOOKEEPER have to be divided by 2! and 3! the reason being that having fixed the positions of the non-repeated letters, the O's may be arranged in 2! ways and the E's in 3! ways without making a different word:
$\frac{9!}{2!3!}=30240$
The answer is: 30,240 ways.